
theorem Th28:
  for L being up-complete Semilattice st for x being Element of L,
E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E
) holds for D being non empty directed Subset of L, x being Element of L holds
  x "/\" sup D = sup ({x} "/\" D)
proof
  let L be up-complete Semilattice such that
A1: for x being Element of L, E being non empty directed Subset of L st
  x <= sup E holds x <= sup ({x} "/\" E);
  let D be non empty directed Subset of L, x be Element of L;
  ex w being Element of L st x >= w & sup D >= w & for c being Element of
  L st x >= c & sup D >= c holds w >= c by LATTICE3:def 11;
  then x "/\" sup D <= sup D by LATTICE3:def 14;
  then
A2: x "/\" sup D <= sup ({x "/\" sup D} "/\" D) by A1;
  reconsider T = {x "/\" sup D} as non empty directed Subset of L by WAYBEL_0:5
;
  ex_sup_of D,L by WAYBEL_0:75;
  then
A3: sup D is_>=_than D by YELLOW_0:30;
  ex_sup_of T "/\" D,L & {x "/\" sup D} "/\" D is_<=_than x "/\" sup D by
WAYBEL_0:75,YELLOW_4:52;
  then sup ({x "/\" sup D} "/\" D) <= x "/\" sup D by YELLOW_0:30;
  hence x "/\" sup D = sup ({x "/\" sup D} "/\" D) by A2,ORDERS_2:2
    .= sup ({x} "/\" {sup D} "/\" D) by YELLOW_4:46
    .= sup ({x} "/\" ({sup D} "/\" D)) by YELLOW_4:41
    .= sup ({x} "/\" D) by A3,YELLOW_4:51;
end;
